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Extremal (in)dependence of a maximum autoregressive process

Marta Ferreira (2013)

Discussiones Mathematicae Probability and Statistics

Maximum autoregressive processes like MARMA (Davis and Resnick, [5] 1989) or power MARMA (Ferreira and Canto e Castro, [12] 2008) have singular joint distributions, an unrealistic feature in most applications. To overcome this pitfall, absolute continuous versions were presented in Alpuim and Athayde [2] (1990) and Ferreira and Canto e Castro [14] (2010b), respectively. We consider an extended version of absolute continuous maximum autoregressive processes that accommodates both asymptotic tail...

Extremes of spheroid shape factor based on two dimensional profiles

Daniel Hlubinka (2006)

Kybernetika

The extremal shape factor of spheroidal particles is studied. Three dimensional particles are considered to be observed via their two dimensional profiles and the problem is to predict the extremal shape factor in a given size class. We proof the stability of the domain of attraction of the spheroid’s and its profile shape factor under a tail equivalence condition. We show namely that the Farlie–Gumbel–Morgenstern bivariate distributions gives the tail uniformity. We provide a way how to find normalising...

Generalized logistic model and its orthant tail dependence

Helena Ferreira, Luisa Pereira (2011)

Kybernetika

The Multivariate Extreme Value distributions have shown their usefulness in environmental studies, financial and insurance mathematics. The Logistic or Gumbel-Hougaard distribution is one of the oldest multivariate extreme value models and it has been extended to asymmetric models. In this paper we introduce generalized logistic multivariate distributions. Our tools are mixtures of copulas and stable mixing variables, extending approaches in Tawn [14], Joe and Hu [6] and Fougères et al. [3]. The...

Heavy tailed durations of regional rainfall

Harry Pavlopoulos, Jan Picek, Jana Jurečková (2008)

Applications of Mathematics

Durations of rain events and drought events over a given region provide important information about the water resources of the region. Of particular interest is the shape of upper tails of the probability distributions of such durations. Recent research suggests that the underlying probability distributions of such durations have heavy tails of hyperbolic type, across a wide range of spatial scales from 2 km to 120 km. These findings are based on radar measurements of spatially averaged rain rate...

Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries

Ludovic Menneteau (2008)

ESAIM: Probability and Statistics

In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.

Multivariate Extreme Value Theory - A Tutorial with Applications to Hydrology and Meteorology

Anne Dutfoy, Sylvie Parey, Nicolas Roche (2014)

Dependence Modeling

In this paper, we provide a tutorial on multivariate extreme value methods which allows to estimate the risk associated with rare events occurring jointly. We draw particular attention to issues related to extremal dependence and we insist on the asymptotic independence feature. We apply the multivariate extreme value theory on two data sets related to hydrology and meteorology: first, the joint flooding of two rivers, which puts at risk the facilities lying downstream the confluence; then the joint...

On characteristic functions of kth record values from the generalized extreme value distribution and its characterization

M. A. W. Mahmoud, M. A. Atallah, M. Albassam (2011)

Applicationes Mathematicae

Recurrence relations for the marginal, joint and conditional characteristic functions of kth record values from the generalized extreme value distribution are established. These relations are utilized to obtain recurrence relations for single, product and conditional moments of kth record values. Moreover, by making use of the recurrence relations the generalized extreme value distribution is characterized.

On the tail dependence in bivariate hydrological frequency analysis

Alexandre Lekina, Fateh Chebana, Taha B. M. J. Ouarda (2015)

Dependence Modeling

In Bivariate Frequency Analysis (BFA) of hydrological events, the study and quantification of the dependence between several variables of interest is commonly carried out through Pearson’s correlation (r), Kendall’s tau (τ) or Spearman’s rho (ρ). These measures provide an overall evaluation of the dependence. However, in BFA, the focus is on the extreme events which occur on the tail of the distribution. Therefore, these measures are not appropriate to quantify the dependence in the tail distribution....

On the tail index estimation of an autoregressive Pareto process

Marta Ferreira (2013)

Discussiones Mathematicae Probability and Statistics

In this paper we consider an autoregressive Pareto process which can be used as an alternative to heavy tailed MARMA. We focus on the tail behavior and prove that the tail empirical quantile function can be approximated by a Gaussian process. This result allows to derive a class of consistent and asymptotically normal estimators for the shape parameter. We will see through simulation that the usual estimation procedure based on an i.i.d. setting may fall short of the desired precision.

On uniform tail expansions of bivariate copulas

Piotr Jaworski (2004)

Applicationes Mathematicae

The theory of copulas provides a useful tool for modelling dependence in risk management. The goal of this paper is to describe the tail behaviour of bivariate copulas and its role in modelling extreme events. We say that a bivariate copula has a uniform lower tail expansion if near the origin it can be approximated by a homogeneous function L(u,v) of degree 1; and it is said to have a uniform upper tail expansion if the associated survival copula has a lower tail expansion. In this paper we (1)...

On uniform tail expansions of multivariate copulas and wide convergence of measures

Piotr Jaworski (2006)

Applicationes Mathematicae

The theory of copulas provides a useful tool for modeling dependence in risk management. In insurance and finance, as well as in other applications, dependence of extreme events is particularly important, hence there is a need for a detailed study of the tail behaviour of multivariate copulas. We investigate the class of copulas having regular tails with a uniform expansion. We present several equivalent characterizations of uniform tail expansions. Next, basing on them, we determine the class of...

Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Antonio Auffinger, Gérard Ben Arous, Sandrine Péché (2009)

Annales de l'I.H.P. Probabilités et statistiques

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in (Electron. Commun. Probab.9 (2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

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